I have questions concerning the definition of the Taylor derivation:
If $I$ is an interval and $x_{0}, x_{0} + \Delta x \in I$ with a $(n + 1)$ differentiable function $f \in C^{n+1} (I, \mathbb{R})$, then we have the Taylor approximation $T_{n}(h)$ of order $n$ given by $$ f(x_{0} + \Delta x) = \sum_{k = 0}^{\infty} \frac{f^{(k)}(x_{0})}{k!}(\Delta x)^{k} + R_{n}) \tag{1}\label{eq1}$$
where $(k)$ is the number of derivative of the function and where the remainder $R_{n}$ satisfies (2) $$ R_{n} = \frac{1}{(n+1)!}f^{(n+1)}(\xi)h^{n+1} \quad \text{for some $\xi$ between $x_{0}$ and $x_{0} + \Delta x$} \tag{2}\label{eq2} $$
Questions:
- in (1), the expression $f^{(k)}$ is the function $f$ to the $k$ derivative? correct?
- in (2) what does the $h$ mean?
- Latex question: how do you numerate the equation on the same line?
Your expression for $f(x_0+\Delta x)$ is not correct. It should be$$\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(\Delta x)^k+R_n,$$that is, it starts with $k=0$ and the $R_n$ term is outside the sum.
Yes, $f^{(k)}$ is the $k$th derivative of $f$. And that $h$ should be $\Delta x$.
Concernin the use of MathJax, read this.